LESSONS IN ALGEBRA. 



Then (o -f d) - (6 + h) = (a - b) + (d - h) ; for each = o + d 



- b- h. 



And (a - d) (b - h) = (a - 1) (d - n> ; for each = a d 



- 6 + k. 



Thus the arithmetical ratio of 11 " 4 i* 7, 

 And the arithmetical ratio of 5 2 is 3. 



Tho ratio of tho um of the term* 10 " 6 IB 10, which is also 

 the sum of tho ratios 7 and 3. 



: atio of the difference of the terms 6 " 2 is 4, which is 



renoo of tho ratios 7 ttii'l :;. 



GEOMETRICAL RATIO w that relation between quantities which 

 it expressed by the QUOTIENT of tlie one divided by tlie ot)w. 



Thus the ratio of 8 to 4 is J or 2 ; fur thin is the quotient of 

 S dividoil l>y 4. In other words, it shows how often 4 is con- 



: in 8. 

 Tho two quantities compared are called a couplet. The first 



the antecedent, and tho last tho consequent. 

 Geometrical ratio is expressed in two ways. 



1. In the form of a fraction, making the antecedent tho 

 numerator, and the consequent the denominator; thus the ratio 



of a to 6 is -. And 



o - 



2. By placing a colon between the quantities compared ; thus, 

 a : b expresses the ratio of a to 6. 



Of those three, the antecedent, the consequent, and the ratio, 

 any two being given, the other may be found. 



Let a = the antecedent, c = the consequent, r = tho ratio. 

 By definition r = a ; that is, the ratio is equal to the 



antecedent divided by the consequent. 

 Multiplying by c, a cr ; that is, the antecedent is equal 

 to the consequent multiplied into tho ratio. 



Dividing by r, c ~ - ; that is, the consequent is equal to the 

 r 



antecedent divided by the ratio. 



If two couplets have their antecedents equal, and their con- 

 sequents equal, their ratios must be equal. 



If in two couplets the ratios are equal, and the antecedents 

 equal, the consequents are equal ; and if the ratios are equal and 

 the consequents equal, the antecedents are equal. 



If tho two quantities compared are equal, the ratio is a unit, 

 or a ratio of equality. The ratio of 3 X 6 : 18 is a unit, for the 

 quotient of any quantity divided by itself is 1. 



If the antecedent of a couplet is greater than the consequent, 

 the ratio is greater than a unit. For if a dividend is greater 

 than its divisor, the quotient is greater than a unit. Thus the 

 ratio of 18:6 is 3. This is called a ratio of greater inequality. 



On the other hand, if the antecedent is less than the con- 

 sequent, the ratio is less than a unit, and is called a ratio of 

 Jess inequality. Thus, the ratio of 2 : 3 is less than a unit, 

 because the dividend is less than the divisor. 



INVERSE or RECIPROCAL RATIO is the ratio of the reciprocals 

 oj two quantities. 



Thus, tho reciprocal ratio of 6 to 3 is g to $ ; that is, J-=-J. 



The direct ratio of a to b is _ ; that is, the antecedent divided 

 by the consequent. . , i i i 7> ;> 



The reciprocal ratio is - : i ; or, --=--=- X - - ; that is, 

 a o a b a i a 



the consequent b divided by the antecedent a. 



Hence a reciprocal ratio is expressed by inverting the fraction 

 which expresses the direct ratio; or when the notation is by 

 points, by inverting the order of the terms. 



Thus, a is to b inversely, as 6 to a. 



COMPOUND RATIO is the ratio of the PRODUCTS of the corre- 

 sponding terms of two or more simple ratios. 



Thus the ratio of 6:3, is 2, 



And the ratio of 12 : 4, is 3. 



The ratio compounded of these is 72 : 12 = 6. 



Here the compound ratio is obtained by multiplying together 

 the two antecedents, and also the two consequents of the simple 

 ratios. Hence it is equal to the product of the simple ratios. 



Compound ratio is not different in its nature from any other 

 ratio. The term is used to denote the origin of the ratio in 

 particular cases. 



If in a Horiea of ratio* the consequent of each preceding 

 coup'et U the antecedent of the following one, the ratio oj lh* 

 first anteo'il: nt I > '. last consequent it equal to that which \t 

 if all the intervening ratiot. 

 Tutu, in the *erie* of ratio*, a : b, 

 b-.c, 

 c : d, 

 d:h, 



the ratio of o : h U equal to that which i* compounded of the 

 ratios of o : 6, of 6 : c, of c . d, and of d : h. For the compound 



ratio by the hut article ia , * = ", or a : h. 

 be dh h 



A particular class of compound ratios is produced by mnlti. 

 plying a simple ratio in itself, or into another equal ratio. The**) 

 are termed duplicate, triplicate, quadruplicate, etc., according 

 to the number of multiplications. 



A ratio compounded of two equal ratios, that is, the s'/uare 

 of the simple ratio, is called a duplicate ratio. 



One compounded of three, that is, the cube of the simple 

 ratio, is called triplicate, etc. 



In a similar manner the ratio of the square roots of two 

 quantities is called a subii uplicate ratio ; that of the cube roott 

 a subtriplicate ratio, etc. 



Thus, the simple ratio of a to b is a : 6. 

 The duplicate ratio of a to 6 is a 2 : b-. 

 The triplicate ratio of a to b is a* : 6*. 

 The subduplicate ratio of a to o is v'a : Vb. 

 The subtriplicate ratio of a to b is \/a . \/b, etc. 



N.B. The terms duplicate, triplicate, etc., must not be con- 

 founded with double, triple, etc. 



The ratio of 6 to 2 is 6:2 = 3. 



Double this ratio, that is, twice the ratio, is 12 : 2 = 6. 

 Triple the ratio, i.e., three times the ratio, is 18 : 2 = 9. 

 The duplicate ratio, i.e., the square of the ratio, is 6 l : 2'- = 9. 

 The triplicate ratio, i.e., the cube of the ratio, is 6 3 : 2 = 27. 



That quantities may have a ratio to each other, it is necea- 

 sary that they should be so far of the same nature, that one can 

 properly be said to be either equal to, or greater, or less than 

 the other. Thus a foot has a ratio to an inch, for one is twelve 

 times as great as the other. 



From the mode of expressing geometrical ratios in the form 

 of a, fraction, it is obvious that the ratio of two quantities is the 

 same as the value of a fraction whose numerator and denomi- 

 nator are equal to the antecedent and consequent of the given 

 ratio. Hence, 



To multiply or divide both the antecedent and consequent by 

 the same quantity, does not alter the ratio. To multiply or 

 divide the antecedent alone by any quantity, multiplies or 

 divides the ratio; to multiply the consequent alone, divides the 

 ratio ; and to divide the consequent, multiplies the ratio. That 

 is, multiplying and dividing the antecedent or consequent has 

 the same effect on the ratio, as a similar operation, performed 

 on the numerator or denominator, has upon the value of a 

 fraction. 



If to or from the terms of any couplet, two other quantities 

 having the same ratio be added or subtracted, the sums or 

 remainders will also have the same ratio. Thus the ratio of 

 12 : 3 is the same as that of 20 : 5. And the ratio of the sum 

 of the antecedents 12 + 20, to the sum of the consequents 3 + 5 

 is the same as the ratio of either couplet. That is, 



12 + 20 : 3 + 5: : 12:3 = 20:5, or 12 + 20 12 20 = 4 



3 + 5 o o 



So also the ratio of the difference of the antecedents to the 

 difference of the consequents is the same. That is, 



20 - 12 : 5 - 3 : : 12 : 3 = 20 : 5, or ^^ = > 2 = ^ = 4 ' 



If in several couplets the ratios are equal, the sum of all the 

 antecedents has the same ratio to the sum of all the consequents, 

 which any one of the antecedents has to its consequent. 



12 : 6 = 2. 



Thus the ratio 



10: 5 = 2. 

 8:4 = 2. 

 6:3 = 2. 



Therefore the ratio of (12 + 10 + 8 +6) : (6 + 5 + 4 + 3) = 2 



